2.21. Plot the following curves, locate any fixed points, and obtain asymp- totic expansions to their solution as k y% +a?, a = Yk (yf+3) 3y+1 (a) Yk+1 = real number, (b) Yk+1 = (c) 2y+1 - 5yk+1Yk + 2y% = 2, ayk+1 (d) Yk+1 = Ykta ? 5y-6yk+2 6y -8yk+3' 5y +6yk+19 Y+5 (е) Ук+1 (f) 5yk+1 (g) Yf+1 = Yk + 6.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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2.8.2
Example B
Consider the following linear difference equation:
3yk+1 = Yk + 2.
(2.177)
This equation has a fixed point yk = 1. The exact solution is given by the
expression
Yk = 1+ A3-k
(2.178)
where A is an arbitrary constant. Consideration of both Figure 2.6 and the
result of equation (2.178) shows that the fixed point is stable.
The linear difference equation
Yk+1 = 2yk – 1
(2.179)
has a fixed point yk = 1 and its exact solution is
Yk = 1+ A2*,
(2.180)
where A is an arbitrary constant. For this case, the fixed point is unstable.
See Figure 2.7.
Likewise, the equation
Yk+1 = -2yk +3
(2.181)
has the fixed point yk = 1. Since the slope is larger in magnitude than one,
the fixed point is unstable. The exact solution is
Yk = 1+ A(-2)*,
(2.182)
where A is an arbitrary constant.
Note that for these three examples, we have, respectively, monotonic con-
vergence, monotonic divergence, and oscillatory divergence. See Figures 2.6,
2.7, and 2.8.
Yk+1
97
(1 + 1)
Yk
FIGURE 2.7: Yk+1 = 2Yk – 1.
Difference Equations
Yk+1
Yk
FIGURE 2.8: Yk+1 = -2yk + 3.
Transcribed Image Text:2.8.2 Example B Consider the following linear difference equation: 3yk+1 = Yk + 2. (2.177) This equation has a fixed point yk = 1. The exact solution is given by the expression Yk = 1+ A3-k (2.178) where A is an arbitrary constant. Consideration of both Figure 2.6 and the result of equation (2.178) shows that the fixed point is stable. The linear difference equation Yk+1 = 2yk – 1 (2.179) has a fixed point yk = 1 and its exact solution is Yk = 1+ A2*, (2.180) where A is an arbitrary constant. For this case, the fixed point is unstable. See Figure 2.7. Likewise, the equation Yk+1 = -2yk +3 (2.181) has the fixed point yk = 1. Since the slope is larger in magnitude than one, the fixed point is unstable. The exact solution is Yk = 1+ A(-2)*, (2.182) where A is an arbitrary constant. Note that for these three examples, we have, respectively, monotonic con- vergence, monotonic divergence, and oscillatory divergence. See Figures 2.6, 2.7, and 2.8. Yk+1 97 (1 + 1) Yk FIGURE 2.7: Yk+1 = 2Yk – 1. Difference Equations Yk+1 Yk FIGURE 2.8: Yk+1 = -2yk + 3.
2.21. Plot the following curves, locate any fixed points, and obtain asymp-
totic expansions to their solution as k
(a) Yk+1 = y + a²,
Yk (Y%+3)
3y%+1
a = real number,
(b) Yk+1 =
(c) 2y+1 - 5yk+1Yk + 2y% = 2,
ayk+1
(d) Yk+1
Yk ta ?
(е) Ук+1
5y% -6yk+2
6y? -8yk+3'
(f) 5yk+1
5y +6yk+19
Y+5
(g) y+1 = Yk +6.
Transcribed Image Text:2.21. Plot the following curves, locate any fixed points, and obtain asymp- totic expansions to their solution as k (a) Yk+1 = y + a², Yk (Y%+3) 3y%+1 a = real number, (b) Yk+1 = (c) 2y+1 - 5yk+1Yk + 2y% = 2, ayk+1 (d) Yk+1 Yk ta ? (е) Ук+1 5y% -6yk+2 6y? -8yk+3' (f) 5yk+1 5y +6yk+19 Y+5 (g) y+1 = Yk +6.
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